Limits of small functors

By Brian Day, Stephen Lack. 2006

74 citations recorded by the source corpus.

Abstract For a small category K enriched over a suitable monoidal category V , the free completion of K under colimits is the presheaf category [ K op , V ] . If K is large, its free completion under colimits is the V -category P K of small presheaves on K , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed structures on P K .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0e60de65541ba4cb9a547fdd825c33a73722ed05

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Citations in the corpus

Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures (2016, background): “Although some aspects of the theory can be developed restricting the attention to small presheaves [19], which support the structure of a pseudomonad,”
A unified framework for generalized multicategories (2009, unknown): “s almost the same as H-Kl(Sq(V-Cat),T). This is not quite right, since T is not really a monad due to size issues. But these problems can be dealt with, for instance by using “small presheaves” as in [DL07]. Assuming the functoriality of H-Klmentioned in Remark 4.4, this observation implies that if ”
Algebraization and Tannaka duality (2014, background): “ian category, and the Yoneda embedding Vect(X) → Vect(d X) is the universal additive map Vect(X) → Ato a cocomplete abelian category. Proof. Left to the reader (see [KS05, Proposition 3.6] as well as [DL07]). Roughly speaking, we view the extra data of the class of surjective maps in Vect(X) as a to”
The basic geometry of Witt vectors, I: The affine case (2008, background): “because their hom-sets are not necessarily true sets, but only Υ-small sets. A more honest approach would be to consider only presheaves and sheaves subject to certain smallness conditions ([41],[17],[14], ...), but since no one has rewritten the foundations of abstract sheaf theory from this point ”
The basic geometry of Witt vectors (2008, background): “A more honest approach would be to consider only presheaves and sheaves subject to certain smallness conditions ([41],[17],[14], .”